Patterson, Bailey - Solid State Physics Introduction to theory

114 3 Electrons in Periodic Potentials

3.1 Reduction to One-Electron Problem

3.1.1 The Variational Principle (B)

The variational principle that will be derived in this Section is often called the Rayleigh–Ritz variational principle. The principle in itself is extremely simple. For this reason, we might be surprised to learn that it is of great practical importance. It gives us a way of constructing energies that have a value greater than or equal to the ground-state energy of the system. In other words, it gives us a way of constructing upper bounds for the energy. There are also techniques for constructing lower bounds for the energy, but these techniques are more complicated and perhaps not so useful.2 The variational technique derived in this Section will be used to derive both the Hartree and Hartree–Fock equations. A variational procedure will also be used with the density functional method to develop the Kohn–Sham equations.

Let H be a positive definite Hermitian operator with eigenvalues Eμ and eigenkets |μ . Since H is positive definite and Hermitian it has a lowest Eμ and the Eμ are real. Let the Eμ be labeled so that E0 is the lowest. Let |ψ be an arbitrary ket (not necessarily normalized) in the space of interest and define a quantity Q(ψ) such that

			Q(ψ) = ψ H ψ .							(3.1)
						ψ ψ
The eigenkets |μ are assumed to form a complete set so that
				ψ		= ∑μ aμ μ .				(3.2)
Since H is Hermitian, we can assume that the |μ are orthonormal, and we find
	ψ ψ	= ∑μ1,μ aμ1 aμ μ1 μ					= ∑μ| aμ |2 ,			(3.3)
and
ψ H ψ		= ∑μ1,μ aμ				1 aμ μ1 H μ = ∑μ | aμ |2 Eμ .				(3.4)
Q can then be written as
Q(ψ) =	∑μ Eμ	| aμ	|2	=	∑μ E0 | aμ |2		+	∑μ (Eμ − E0 ) |	aμ |2	,
	∑μ| aμ |2					∑μ| aμ |2		∑μ| aμ |2

2 See, for example, Friedman [3.18].

	3.1 Reduction to One-Electron Problem			115
or
Q(ψ) = E0 +	∑μ (Eμ − E0 ) | aμ |2		.	(3.5)
		∑μ | aμ |2
Since Eμ > E0 and |aμ|2 ≥ 0, we can immediately conclude from (3.5) that
Q(ψ) ≥ E0 .				(3.6)
Summarizing, we have
ψ H ψ		≥ E0 .		(3.7)
ψ ψ

Equation (3.7) is the basic equation of the variational principle. Suppose ψ is a trial wave function with a variable parameter η. Then the η that are the best if Q(ψ) is to be as close to the lowest eigenvalue as possible (or as close to the ground-state energy if H is the Hamiltonian) are among the η for which

	∂Q	= 0 .	(3.8)
	∂η

For the η = ηb that solves (3.8) and minimizes Q(ψ), Q(ψ(ηb)) is an approximation to E0. By using successively more sophisticated trial wave functions with more and more variable parameters (this is where the hard work comes in), we can get as close to E0 as desired. Q(ψ) = E0 exactly only if ψ is an exact wave function corresponding to E0.

3.1.2 The Hartree Approximation (B)

When applied to electrons, the Hartree method neglects the effects of antisymmetry of many electron wave functions. It also neglects correlations (this term will be defined precisely later). Despite these deficiencies, the Hartree approximation can be very useful, e.g. when applied to many-electron atoms. The fact that we have a shell structure in atoms appears to make the deficiencies of the Hartree approximation not very serious (strictly speaking even here we have to use some of the ideas of the Pauli principle in order that all electrons are not in the same lowest-energy shell). The Hartree approximation is also useful for gaining a crude understanding of why the quasifree-electron picture of metals has some validity. Finally, it is easier to understand the Hartree–Fock method as well as the density functional method by slowly building up the requisite ideas. The Hartree approximation is a first step.

116 3 Electrons in Periodic Potentials

For a solid, the many-electron Hamiltonian whose Schrödinger wave equation must be solved is

2

H= − 2m ∑i

+12 ∑a′,b

(electrons) i2 − ∑a (nuclei)					e2
					4πε	0	r
		i (electrons)					ai				(3.9)
	ZaZbe2								e2
(nuclei)		+	1	∑ ′						.
	4πε0Rab	2		i, j (electron)				4πε0rij

This equals H0 of (2.10).

The first term in the Hamiltonian is the operator representing the kinetic energy of all the electrons. Each different i corresponds to a different electron The second term is the potential energy of interaction of all of the electrons with all of the nuclei, and rai is the distance from the ath nucleus to the ith electron. This potential energy of interaction is due to the Coulomb forces. Za is the atomic number of the nucleus at a. The third term is the Coulomb potential energy of interaction between the nuclei. Rab is the distance between nucleus a and nucleus b. The prime on the sum as usual means omission of those terms for which a = b. The fourth term is the Coulomb potential energy of interaction between the electrons, and rij is the distance between the ith and jth electrons. For electronic calculations, the internuclear distances are treated as constant parameters, and so the third term can be omitted. This is in accord with the Born–Oppenheimer approximation as discussed at the beginning of Chap. 2. Magnetic interactions are relativistic corrections to the electrical interactions, and so are often small. They are omitted in (3.9).

For the purpose of deriving the Hartree approximation, this N-electron Hamiltonian is unnecessarily cumbersome. It is more convenient to write it in the more abstract form

H (x1 xn ) = ∑iN=1H (i) +	1	∑i,′ j V (ij) ,	(3.10a)
	2
where
V (ij) =V ( ji) .			(3.10b)

In (3.10a), H(i) is a one-particle operator (e.g. the kinetic energy), V(ij) is a twoparticle operator (e.g. the fourth term in (3.9)), and i refers to the electron with coordinate xi (or ri if you prefer). Spin does not need to be discussed for a while, but again we can regard xi in a wave function as including the spin of electron i if we so desire.

Eigenfunctions of the many-electron Hamiltonian defined by (3.10a) will be sought by use of the variational principle. If there were no interaction between electrons and if the indistinguishability of electrons is forgotten, then the eigenfunction can be a product of N functions, each function being a function of

3.1 Reduction to One-Electron Problem	117

the coordinates of only one electron. So even though we have interactions, let us try a trial wave function that is a simple product of one-electron wave functions:

ψ(x1 xn ) = u1(x1)u2 (x2 ) un (xn ) .

(3.11)

The u will be assumed to be normalized, but not necessarily orthogonal. Since the u are normalized, it is easy to show that the ψ are normalized:

∫ψ (x1, , xN )ψ(x1, , xN )dτ = ∫ u1 (x1)u(x1)dτ1 ∫ uN (xN )u(xN )dτN =1.

Combining (3.10) and (3.11), we can easily calculate

ψ H ψ ≡ ∫ψ Hψdτ

	= ∫u1 (x1) uN (xN )[∑H (i) + 12 ∑i,′ j V(ij)]u1(x1) uN (xN )dτ
	= ∑i ∫ui (xi )H (i)ui (xi )dτi	(3.12)
	+ 12 ∑i,′ j ∫ui (xi )u j (x j )V(ij)ui (xi )u j (x j )dτidτ j
	= ∑i ∫ui (x1)H (1)ui (x1)dτ1

+ 12 ∑i,′ j ∫ui (x1)u j (x2 )V(1,2)ui (x1)u j (x2 )dτ1dτ2 ,

where the last equation comes from making changes of dummy integration variables.

By (3.7) we need to find an extremum (hopefully a minimum) for ψ|H|ψ while at the same time taking into account the constraint of normalization. The convenient way to do this is by the use of Lagrange multipliers [2]. The variational principle then tells us that the best choice of u is determined from

δ[ ψ H ψ − ∑i λi ∫ ui (xi )ui (xi )dτi ] = 0 .

(3.13)

In (3.13), δ is an arbitrary variation of the u. ui and uj can be treated independently (since Lagrange multipliers λi are being used) as can ui and uj*. Thus it is

convenient to choose δ = δk, where δkuk* and δkuk are independent and arbitrary, δkui (≠k) = 0, and δkui*(≠k) = 0.

By (3.10b), (3.12), (3.13), δ = δk, and a little manipulation we easily find

∫δ u (x1){[H (1)u (x1) + (∑ j ( k) ∫ u (x2 )V (1,2)u (x2 )dτ)u (x1)] (3.14) k k k ≠ j j k

− λkuk (x1)}dτ + C.C. = 0.

In (3.14), C.C. means the complex conjugate of the terms that have already been written on the left-hand side of (3.14). The second term is easily seen to be the complex conjugate of the first term because

δ ψ H ψ = δψ H ψ + ψ H δψ = δψ H ψ + δψ H ψ , since H is Hermitian.

118 3 Electrons in Periodic Potentials

In (3.14), two terms have been combined by making changes of dummy summation and integration variables, and by using the fact that V(1,2) = V(2,1). In (3.14), δkuk*(x1) and δkuk(x1) are independent and arbitrary, so that the integrands involved in the coefficients of either δkuk or δkuk* must be zero. The latter fact gives the Hartree equations

H (x1)uk (x1) +[∑ j (≠k) ∫u j (x2 )V (1,2)u j (x2 )dτ2 ]uk (x1) = λk uk (x1) . (3.15)

Because we will have to do the same sort of manipulation when we derive the Hartree–Fock equations, we will add a few comments on the derivation of (3.15). Allowing for the possibility that the λk may be complex, the most general form of (3.14) is

∫δkuk (x1){F(1)uk (1) − λkuk (x1)}dτ1

+∫δkuk (x1){F(1)uk (1) − λkuk (x1)} dτ1 = 0,

where F(1) is defined by (3.14). Since δkuk(x1) and δkuk(x1)* are independent (which we will argue in a moment), we have

F (1)uk (1) = λk uk (1) and F (1)uk (1) = λk uk (1) .

F is Hermitian so that these equations are consistent because then λk = λk* and is real. The independence of δkuk and δkuk* is easily seen by the fact that if δkuk = α + iβ then α and β are real and independent. Therefore if

(C + C	2	)α + (C − C	2	)iβ = 0, then	C = C	2	and C = −C	2	,
1		1			1		1

or Cl = C2 = 0 because this is what we mean by independence. But this implies Cl(α + iβ) + C2(α − iβ) = 0 implies Cl = C2 = 0 so α + iβ = δkuk and α − iβ = δkuk* are independent.

Several comments can be made about these equations. The Hartree approximation takes us from one Schrödinger equation for N electrons to N Schrödinger equations each for one electron. The way to solve the Hartree equations is to guess a set of ui and then use (3.15) to calculate a new set. This process is to be continued until the u we calculate are similar to the u we guess. When this stage is reached, we say we have a consistent set of equations. In the Hartree approximation, the state ui is not determined by the instantaneous positions of the electrons in state j, but only by their average positions. That is, the sum −e ∑j(≠k)uj*(x2)uj(x2) serves as a time-independent density ρ(2) of electrons for calculating uk(x1). If V(1,2) is the Coulomb repulsion between electrons, the second term on the left-hand side corresponds to

	− ∫ ρ(2)	1	dτ2 .
		4πε0r12

Thus this term has a classical and intuitive meaning. The ui, obtained by solving the Hartree equations in a self-consistent manner, are the best set of one-electron orbitals in the sense that for these orbitals Q(ψ) = ψ|H|ψ / ψ|ψ (with ψ = ul,…,uN) is a minimum. The physical interpretation of the Lagrange multipliers λk has not yet

3.1 Reduction to One-Electron Problem	119

been given. Their values are determined by the eigenvalue condition as expressed by (3.15). From the form of the Hartree equations we might expect that the λk correspond to “the energy of an electron in state k.” This will be further discussed and made precise within the more general context of the Hartree–Fock approximation.

3.1.3 The Hartree–Fock Approximation (A)

The derivation of the Hartree–Fock equations is similar to the derivation of the Hartree equations. The difference in the two methods lies in the form of the trial wave function that is used. In the Hartree–Fock approximation the fact that electrons are fermions and must have antisymmetric wave functions is explicitly taken into account. If we introduce a “spin coordinate” for each electron, and let this spin coordinate take on two possible values (say ± ½), then the general way we put into the Pauli principle is to require that the many-particle wave function be antisymmetric in the interchange of all the coordinates of any two electrons. If we form the antisymmetric many-particle wave functions out of one-particle wave functions, then we are led to the idea of the Slater determinant for the trial wave function. Applying the ideas of the variational principle, we are then led to the Hartree–Fock equations. The details of this program are given below. First, we shall derive the Hartree–Fock equations using the same notation as was used for the Hartree equations. We will then repeat the derivation using the more convenient second quantization notation. The second quantization notation often shortens the algebra of such derivations. Since much of the current literature is presented in the second quantization notation, some familiarity with this method is necessary.

Derivation of Hartree–Fock Equations in Old Notation (A)3

Given N one-particle wave functions ui(xi), where xi in the wave functions represents all the coordinates (space and spin) of particle i, there is only one antisymmetric combination that can be formed (this is a theorem that we will not prove). This antisymmetric combination is a determinant. Thus the trial wave function that will be used takes the form

	u1(x1)	u2 (x1)	uN (x1)
ψ(x1,…, xN ) = M	u1(x2 )	u2 (x2 )	uN (x2 )	.	(3.16)
	u1(xN ) u2 (xN )		uN (xN )

In (3.16), M is a normalizing factor to be chosen so that ∫|ψ|2dτ = 1.

3Actually, for the most part we assume restricted Hartree–Fock Equations where there are an even number of electrons divided into sets of 2 with the same spatial wave functions paired with either a spin-up or spin-down function. In unrestricted Hartree–Fock we do not make these assumptions. See, e.g., Marder [3.34, p. 209].

120 3 Electrons in Periodic Potentials

It is easy to see why the use of a determinant automatically takes into account the Pauli principle. If two electrons are in the same state, then for some i and j, ui = uj. But then two columns of the determinant would be equal and hence ψ = 0, or in other words ui = uj is physically impossible. For the same reason, two electrons with the same spin cannot occupy the same point in space. The antisymmetry property is also easy to see. If we interchange xi and xj, then two rows of the determinant are interchanged so that ψ changes sign. All physical properties of the system in state ψ depend only quadratically on ψ, so the physical properties are unaffected by the change of sign caused by the interchange of the two electrons. This is an example of the indistinguishability of electrons. Rather than using (3.16) directly, it is more convenient to write the determinant in terms of its definition that uses permutation operators:

ψ(x1 xn ) = M ∑ p (−) p Pu1(x1) uN (xN ) .

(3.17)

In (3.17), P is the permutation operator and it acts either on the subscripts of u (in pairs) or on the coordinates xi (in pairs). (−)P is ±1, depending on whether P is an even or an odd permutation. A permutation of a set is even (odd), if it takes an even (odd) number of interchanges of pairs of the set to get the set from its original order to its permuted order.

In (3.17) it will be assumed that the single-particle wave functions are

orthonormal:
∫ ui (x1)u j (x1)dx1 = δij .	(3.18)

In (3.18) the symbol ∫ means to integrate over the spatial coordinates and to sum over the spin coordinates. For the purposes of this calculation, however, the symbol can be regarded as an ordinary integral (most of the time) and things will come out satisfactorily.

From Problem 3.2, the correct normalizing factor for the ψ is (N!)−1/2, and so the normalized ψ have the form

ψ(x1 xn ) = (1/ N !)∑ p (−) p Pu1(x1) uN (xN ) .

(3.19)

Functions of the form (3.19) are called Slater determinants.

The next obvious step is to apply the variational principle. Using Lagrange multipliers λij to take into account the orthonormality constraint, we have

δ (ψ H ψ − ∑i, j λi, j ui u j )= 0 .

(3.20)

Using the same Hamiltonian as was used in the Hartree problem, we have

ψ H ψ = ψ ∑ H (i) ψ + ψ	1	∑ ′	V (ij) ψ .	(3.21)
	2	i, j

3.1 Reduction to One-Electron Problem	121

The first term can be evaluated as follows:

ψ ∑H (i)ψ

=	1	∑ p, p′(−) p+ p′∫[Pu1 (x1)			uN (xN )]∑H (i)[P′u1(x1)	uN (xN )]dτ
	N!
=	1	∑ p, p′(−) p+ p′ P∫[u1 (x1)			uN (xN )]∑H (i)P−1P′[u1(x1) uN (xN )]dτ,
	N!
since P commutes with ∑ H(i). Defining Q = P−1P′, we have
ψ ∑ H (i) ψ
	=	1	∑ p,q (−)q P∫[u1	(x1)	uN (xN )]∑ H (i)Q[u1(x1)	uN (xN )]dτ,
		N!
where Q ≡ P−1P′ is also a permutation,
		= ∑q (−)q ∫[u1 (x1)		uN (xN )]∑ H (i)Q[u1(x1) uN (xN )]dτ,

where P is regarded as acting on the coordinates, and by dummy changes of integration variables, the N! integrals are identical,

= ∑	q	(−)q ∫[u1 (x1)	uN (xN )]∑ H (i)[uq (x1)	uq	N	(xN )]dτ,
			1

where ql…qN is the permutation of 1…N generated by Q,

= ∑q (−)q ∑i ∫ ui H (i)uqi δq1	δq2	δqi−1	δqi+1	δqN	dτi ,
1	2	i −1	i +1	N
where use has been made of the orthonormality of the ui,
= ∑i ∫ ui (x1)H (1)u1(x1)dτ1 ,					(3.22)

where the delta functions allow only Q = I (the identity) and a dummy change of integration variables has been made.

The derivation of an expression for the matrix element of the two-particle operator is somewhat longer:

ψ	1		∑ ′		V (i, j) ψ
		2	i, j
	=		1	∑ p, p′(−) p+ p′∫[Pu1 (x1)		uN (xN )]
		2N!			×∑i,′ j V (i, j)[P′u1(x1) uN (xN )]dτ
	=		1	∑ p, p′(−) p+ p′ P{∫[u1 (x1)		uN (xN )]
		2N!			×∑i,′ j V (i, j)P−1P′[u1(x1) uN (xN )]dτ},

122 3 Electrons in Periodic Potentials

since P commutes with ∑′i,jV(i,j),

=

1

∑ p,q (−)q P[∫u1 (x1) uN (xN )∑i,′ jV (i, j)Qu1(x1) uN (xN )dτ],

2N!

where Q ≡ P−1P′ is also a permutation,

=

1

∑

q

(−)q ∫[u1 (x1)

uN (xN )]∑ ′

V (i, j)[uq (x1)

uq

(xN )]dτ ,

2N!

N

i, j

1

since all N! integrals generated by P can be shown to be identical and q1…qN is the permutation of 1…N generated by Q,

=

1

∑q (−)q ∑i,′ j ∫ ui (xi )u j (x j )V (i, j)uqi

(xi )uq j (x j )dτidτ jδq1

δqi−1

2

δ j −1 δ

1

i −1

×δ i+1

j+1

δ

N

,

qi +1

q j −1

q j +1

qN

where use has been made of the orthonormality of the ui,

=

1

∑i,′ j ∫[ui (x1)u j (x2 )V (1,2)ui (x1)u j

(x2 )

(3.23)

2

− u (x )u

(x

)V (1,2)u

(x )u

(x

)]dτ dτ

,

2

j

i

2

2

i 1 j

1

1

where the delta function allows only qi = i, qj = j or qi = j, qj = i, and these permutations differ in the sign of (−1)q and a change in the dummy variables of integration has been made.

Combining (3.20), (3.21), (3.22), (3.23), and choosing δ = δk in the same way as was done in the Hartree approximation, we find

∫dτ1δk uk (x1){H (1)uk (x1) + ∑ j(≠k) ∫dτ2u j (x2 )V (1,2)u j (x2 )uk (x2 )

−∑ j(≠k) ∫dτ2u j (x2 )V (1,2)uk (x2 )u j (x1) − ∑ j u j (x1)λkj }

+C.C. = 0.

Since δkuk* is completely arbitrary, the part of the integrand inside the brackets must vanish. There is some arbitrariness in the λ just because the u are not unique (there are several sets of us that yield the same determinant). The arbitrariness is

sufficient that we can choose λk≠j can let the sums run over j = k as equations are thus obtained:

= 0 without loss in generality. Also note that we the j = k terms cancel one another. The following

H (1)u (x ) + ∑ [∫dτ u (x )V (1,2)u (x )u (x )

k 1 j 2 j 2 j 2 k 1 (3.24) − ∫dτ2u j (x2 )V (1,2)uk (x2 )u j (x1)] = εk uk ,

where εk ≡ λkk.

3.1 Reduction to One-Electron Problem	123

Equation (3.24) gives the set of equations known as the Hartree–Fock equations. The derivation is not complete until the εk are interpreted. From (3.24) we can write

εk = uk (1) H (1) uk (1) + ∑ j { uk (1)u j (2) V (1,2) uk (1)u j (2)	},	(3.25)
− uk (1)u j (2) V (1,2) u j (1)uk (2)

where 1 and 2 are a notation for x1 and x2. It is convenient at this point to be explicit about what we mean by this notation. We must realize that

uk (x1) ≡ψk (r1)ξk (s1) ,

(3.26)

where ψk is the spatial part of the wave function, and ξk is the spin part.

Integrals mean integration over space and summation over spins. The spin functions refer to either “+1/2” or “−1/2” spin states, where ±1/2 refers to the eigenvalues of sz/ for the spin in question. Two spin functions have inner product equal to one when they are both in the same spin state. They have inner product equal to zero when one is in a +1/2 spin state and one is in a −1/2 spin state. Let us rewrite (3.25) where the summation over the spin part of the inner product has already been done. The inner products now refer only to integration over space:

εk = ψ k (1) H (1) ψ k (1) + ∑ j ψ k (1)ψ j (2) V (1,2) ψ k (1)ψ j (2) (3.27) −∑j(||k) ψ k (1)ψ j (2) V (1,2) ψ j (1)ψ k (2) .

In (3.27), j(|| k) means to sum only over states j that have spins that are in the same state as those states labeled by k.

Equation (3.27), of course, does not tell us what the εk are. A theorem due to Koopmans gives the desired interpretation. Koopmans’ theorem states that εk is the negative of the energy required to remove an electron in state k from the solid. The proof is fairly simple. From (3.22) and (3.23) we can write (using the same notation as in (3.27))

E = ∑i ψi (1) H (1)ψi (1) +	1	∑i, j		ψi (1)ψ j (2) V (1,2)				ψi (1)ψ j			(2)
− 1	2			ψ	(1)ψ		(2) V (1,2)	ψ		(1)ψ	(3.28)
	∑		i,j(||)			j			j		(2) .
2				i						i

Denoting E(w.o.k.) as (3.28) in which terms for which i = k, j = k are omitted from the sums we have

E(w.o.k.) − E = − ψk (1) H (1) ψk (1)
− ∑ j ψk (1)ψ j (2) V (1,2) ψk (1)ψ j (2)	(3.29)
+ ∑i, j(||) ψk (1)ψ j (2) V (1,2) ψ j (1)ψk (2) .
Combining (3.27) and (3.29), we have
εk = −[E(w.o.k.) − E] ,	(3.30)